Extruding parts of a complex mesh
If, in 2.79. you hit CtrlAltU, you can bring up User Preferences. In the Add-Ons tab, you will find 'Add Mesh: Extra Objects', shipped with the application. Activate it, and save the settings.. it will enable you to create an object like this:


.. which might be a good starting point for your object. Note not all the faces are hexagons. ( You can't tessellate the surface of a sphere with regular hexagons, there need to be 'pleats': here, intrusions of pentagonal faces)

If you have surplus triangulation, you can use X > Limited Dissolve to remove coplanar edges.

If you delete the pentagonal faces from a copy of the base object, you can use the Duplifaces mechanism for distributing your hexagonal object onto the remaining hexagonal faces. (If the pentagons are acceptable, you could repeat the process to distribute a pentagonal sculpted object.)



Throughout modeling, you may find ShiftG > Select Similar, is a good way to select sets of elements across the surface of the base object.